Problem: $ 0.\overline{68} \div 2.\overline{4} = {?} $
Solution: First convert the repeating decimals to fractions. $\begin{align*} 100x &= 68.6868...\\ x &= 0.6868...\end{align*} $ $\begin{align*} 99x &= 68 \\ x &= \dfrac{68}{99}\end{align*} $ $\begin{align*} 10y &= 24.4444...\\ y &= 2.4444...\end{align*} $ $\begin{align*} 9y &= 22 \\ y &= \dfrac{22}{9}\end{align*} $ So, the problem becomes: $ \dfrac{68}{99} \div \dfrac{22}{9} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ \dfrac{68}{99} \times \dfrac{9}{22} = {?} $ $ \phantom{\dfrac{68}{99} \times \dfrac{22}{9}} = \dfrac{68 \times 9}{99 \times 22} $ $ \phantom{\dfrac{68}{99} \times \dfrac{22}{9}} = \dfrac{68 \times \cancel{9}} {\cancel{99}11 \times 22} $ $ \phantom{\dfrac{68}{99} \times \dfrac{22}{9}} = \dfrac{68}{242} $ Simplify: ${= \dfrac{34}{121}}$